In the most general form, the dynamics of a quantum-mechanical system are determined by a master equation in Hilbert space: an equation of motion for the density operator (usually written ) of the system. The density operator is defined with respect to a completeorthonormal basis. Although it is possible to directly integrate this equation for very small systems (i.e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove that the density can always be written in a diagonal form, provided that it is with respect to an overcomplete basis. When the density operator is represented in such an overcomplete basis, then it can be written in a way more like an ordinary function, at the expense that the function has the features of a quasiprobability distribution. The evolution of the system is then completely determined by the evolution of the quasiprobability distribution function.
They also have some additional interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then
Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof.
Proof of the overcompleteness of the coherent states
The function f is not unique. There exists a family of different representations, each connected to a different ordering Ω. The most popular in the general physics literature and historically first of these is the Wigner quasiprobability distribution, which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:
If the quantum system has a classical analog, e.g. a coherent state or thermal radiation, then P is non-negative everywhere like an ordinary probability distribution. If, however, the quantum system has no classical analog, e.g. an incoherent Fock state or entangled system, then P is negative somewhere or more singular than a delta function.
This sweeping statement is unavailable in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog.
In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti-normal order. More recently, the positive P representation and a wider class of generalized P representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function.
Analogous to probability theory, quantum quasiprobability distributions
can be written in terms of characteristic functions,
from which all operator expectation values can be derived. The characteristic
functions for the Wigner, Glauber P and Q distributions of an N mode system
are as follows:
Here and are vectors containing the annihilation and creation operators for each mode
of the system. These characteristic functions can be used to directly evaluate expectation values of operator moments. The ordering of the annihilation and creation operators in these moments is specific to the particular characteristic function. For instance, normally ordered (annihilation operators preceding creation operators) moments can be evaluated in the following way from :
In the same way, expectation values of anti-normally ordered and symmetrically ordered combinations of annihilation and creation operators can be evaluated from the characteristic functions for the Q and Wigner distributions, respectively. The quasiprobability functions themselves are defined as Fourier transforms of the above characteristic functions. That is,
Here and may be identified as coherent state amplitudes in the case of the Glauber P and Q distributions, but simply c-numbers for the Wigner function. Since differentiation in normal space becomes multiplication in Fourier space, moments can be calculated from these functions in the following way:
Since each of the above transformations from ρ to the distribution functions is linear, the equation of motion for each distribution can be obtained by performing the same transformations to . Furthermore, as any master equation which can be expressed in Lindblad form is completely described by the action of combinations of annihilation and creation operators on the density operator, it is useful to consider the effect such operations have on each of the quasiprobability functions.
For instance, consider the annihilation operator acting on ρ. For the characteristic function of the P distribution we have
Taking the Fourier transform with respect to to find the
action corresponding action on the Glauber P function, we find
By following this procedure for each of the above distributions, the following
operator correspondences can be identified:
Here κ = 0, 1/2 or 1 for P, Wigner, and Q distributions, respectively. In this way, master equations can be expressed as an equations of
motion of quasiprobability functions.
Since for n>0 this is more singular than a delta function, a Fock state has no classical analog. The non-classicality is less transparent as one proceeds with the Gaussian convolutions. If Ln is the nth Laguerre polynomial, W is
which can go negative but is bounded. Q always remains positive and bounded: